Introducing congruent triangles

Last class was devoted to introducing and discussing congruent shapes. I started the introduction by asking students to draw a simple figure, perhaps using a straight edge of graph paper. I then asked the class to make an exact copy of the figure they drew.

With these in hand, I asked the class how they knew that the figures were exactly the same? This led to a discussion about characteristics that each figure needed. The class boiled it down to each figure having the same number of sides with matching side lengths and that all the angles had to match in measurement.

I then tied this back to rigid motions. With rigid motions we are creating a pre-image and image that are congruent. I referenced how we labeled both images and then proceeded to show different pre-image and image figures from rigid motions placed on a graph.

The task for the class was how to mark the figures to show which pieces were congruent. There was a bit of hesitancy at first, but a student came to the board and appropriately marked the congruent sides. Another student then marked the congruent angles. This happened with the second pair of figures as well.

The third pair of figures presented a slight wrinkle. As before, a student marked congruent sides and then congruent angles. The pair presented were isosceles triangles. As the rest of the class looked at the markings, a couple of hands shot up. One girl asked whether two sides should actually be marked congruent. Others in the class agreed, noting that the triangle was isosceles and should have two congruent sides. (They were using actual side length to determine the congruence.) Then another student said that the base angles should also be marked as congruent.

I was pleased with students recognizing the anomaly in the last pair. I was also pleased with how comfortable students were with marking congruent sides and congruent angles.

I mentioned that what they had identified was that corresponding parts of congruent triangles are congruent. This was the first mention of what is commonly abbreviated CPCTC. I didn't dwell on this, but wanted to point it out in case they ran across this terminology.

I gave students some practice problems, identifying corresponding parts or writing congruence statements. I used the practice sheet problems that showed congruence statements as the guide of how to write a congruence statement. They readily picked it up and seemed comfortable working through the problems.

I next started the thinking about how much information is needed to convey congruence between triangles. I used a tweeting triangle lesson that I have used over the years.

The tweeting triangles lesson outline is posted on my website. I'll have students try their ideas next class and we'll, hopefully, build all of the triangle congruence shortcuts.